The Heavens in a Pile of Sand

Science  01 Nov 2002:
Vol. 298, Issue 5595, pp. 979-980
DOI: 10.1126/science.1075555

Dramatic auroral displays over Earth's polar regions reveal a range of intricate patterns. Auroral spirals, westward-traveling surges, auroral curls, and many other phenomena have been identified on time scales from seconds to hours and spatial scales from 100 meters to thousands of kilometers (see the figure, panel A) (1). Recently, attention has begun to focus on whether there may be universal aspects to auroral structure, of the type observed in many complex natural systems (2).

One universal footprint seen in many complex systems is self-affinity. An object is self-affine if it is the same when viewed on any scale. In the early 1990s, the self-affinity of the aurora began to be noticed (3). Takalo et al. investigated this by analyzing the auroral electrojet (AE) index—a time series of the peak magnetic perturbation on the ground caused by electrical currents in the aurora, 100 km overhead (4). They found that the average squared difference between equally spaced points of the AE index time series doubled when the spacing was quadrupled, for any resolution between 1 and 100 min (4).

This fractional power law relationship is a classic definition of a self-affine structure—a “fractal.” The fractal behavior was broken for time scales longer than 100 min, a feature thought to correspond to sporadic interruptions of the time series by a global auroral disturbance known as the substorm. Other measures of the AE index were also found to have power law relationships. For example, the probability distribution of the time for which the AE index exceeded a given fixed threshold followed a power law distribution from 1 to 1000 min. Superposed on the power law was another distribution centered on a fixed scale of ∼100 min, corresponding to the substorm (5, 6, 7).

A similar measure was used to investigate the spatial structure of the aurora. In ultraviolet images of the aurora from NASA's Polar spacecraft, Lui et al. (8) identified auroral bright spots where the auroral emission intensity exceeded some fixed threshold, during both quiet and substorm intervals. They found a power law relationship between the number of bright spots and their area between 3000 and 1 million km2. An additional population was found to be centered on a fixed scale of ∼2 million km2, corresponding to the substorm disturbance.

Because auroral bright spots evolve in space and time, a bright spot counted in one image at one time may be the same as another bright spot counted at an earlier or later time. Thus, the Lui et al. method overestimates the number of evolving spots. Uritsky et al. (9) reanalyzed Polar spacecraft auroral images, taking this spatiotemporal evolution into account (see the figure, panel B). The probability distribution of maximum bright spot area, and of area integrated over bright spot lifetime, now both followed a power law distribution over the entire observable range (three to five orders of magnitude). The probability distributions of bright spot lifetime, maximum dissipated power, and dissipated energy also followed power laws.

From these [and other (57)] observations, a picture has emerged of dynamically evolving “avalanches” of bright aurora with fractal structure in time and space. But why does nature paint such a picture? What is the answer in the auroral context to Kadanoff's famous question: “Fractals: Where's the physics?” (10).

Complex beauty.

(A) Ultraviolet image of Earth's aurora viewed from space by NASA's Polar satellite [21:19:23 universal time (UT), 16 February 1997]. Structures in the auroral luminosity are seen on many scales from the global to the image resolution (∼20 km) and evolve on many different time scales. (B) When ∼12,300 different evolving auroral spots above a given luminosity threshold were tracked (9), the number of bright spots of given maximum area followed a power law (a straight line when plotted with logarithmic axes). This means that there is no typical bright spot size. (C) An optical image of the aurora taken from the ground at about the same time (21:20:20 UT) as (A) reveals details of auroral structures to even smaller scales. The image radius is ∼200 km centered above Kilpisjärvi, Finland. North is at the top of the image; the bright round spot in the southwest sky is the Moon.


Bak et al. (11) proposed that scale invariance in nature might be a result of self-organized criticality (SOC)—the tendency of slowly driven, interaction-dominated, thresholded systems to self-organize to a critical state, independent of the initial conditions. Here, the critical state is a statistically steady but nonequilibrium state in which the accumulating energy from the driver is released spasmodically in “avalanches” with a self-similar size distribution. This state arises from long-range correlations established through short-range interactions in a system with many degrees of freedom (12). SOC was first identified in computer experiments mimicking a slowly growing pile of sand and was experimentally established in a pile of rice (13).

The SOC paradigm has been applied to many natural systems, from earthquakes, measles, and forest fires to astrophysical accretion disks (14). Contemporary with the experimental work of Takalo et al., several theorists recognized that criticality may offer an explanation for self-affine auroral structure (5, 6). But why should the aurora behave like a sand pile (6)?

The aurora can be likened to a giant natural television screen. Charged particles from a natural electron gun in space are guided by Earth's magnetic field toward the polar atmosphere, where they collide with other particles and give off light. Thus, the aurora is a projection of the dynamic charged particle structure of the near-Earth space that is magnetically connected to Earth (a region known as the magnetosphere).

The collective dynamics of charged particles in a magnetic field is described by the equations of magnetohydrodynamics (MHD). Recently it has been shown that the MHD equations can be mapped onto discrete SOC equations (15). The scale-free structure of the aurora is then argued to come from a scale-free structure of a SOC magnetosphere. Indeed, satellite measurements in the magnetosphere have begun to show preliminary evidence of SOC in that fast flows of charged particles correlated with auroral emissions have a scale-free distribution of durations like that of SOC (16).

However, the scale-free structure of the aurora may not come from scale-free structure in the magnetosphere. Instead, it may arise from a similar scale-free structure in the turbulent solar wind that drives the magnetosphere (7). Studies using long (but non-overlapping) solar wind and auroral time series show the same static fractal properties, but comparisons using shorter but overlapping series could indicate that this may be coincidental rather than causal (17). Work continues on this issue, which is an example of a generic problem of complex systems coupled to complex drivers.

Another question, emphasized by Consolini and Chang (5), is how well the assumptions of the SOC model are met in the magnetosphere. In the original SOC models, scale-free behavior only emerged when the driving rate was very slow compared to the interaction time scales. However, another kind of nonequilibrium system exists in which the scale-free behavior only appears when the driving rate becomes sufficiently fast (18) (an example of forced criticality).

The debate between a driven and an internal origin for intermittent scale-free dynamics is not at all unique to the magnetosphere. It parallels a debate in theories of punctuated evolution between the influence of “external” events (such as asteroid impact) on extinctions and self-organized “internal” extinctions (19). Thus, complexity is providing a new approach to addressing long-standing space science problems, while in return space science is beginning to play an active role in addressing fundamental topical issues in complexity.

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